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Sharp multiplicative inequalities with $\mathrm{BMO}$ $\mathrm{II}$
Classical Analysis and ODEs · Submitted 10 Nov 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We find the best possible constant $C$ in the inequality $$\|\varphi\|_{L^r}^{\phantom{\frac{p}{r}}}\leq C\|\varphi\|_{L^p}^{\frac{p}{r}}\|\varphi\|_{\mathrm{BMO}}^{1-\frac{p}{r}}$$ for all possible values of parameters $p$ and $r$ such that $1 \le p < r < +\infty$.
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The abstract, sentence by sentence
We find the best possible constant $C$ in the inequality $$\|\varphi\|_{L^r}^{\phantom{\frac{p}{r}}}\leq C\|\varphi\|_{L^p}^{\frac{p}{r}}\|\varphi\|_{\mathrm{BMO}}^{1-\frac{p}{r}}$$ for all possible values of parameters $p$ and $r$ such that $1 \le p < r < +\infty$. We employ the Bellman function technique to solve this problem. The Bellman function of three variables corresponding to this problem has a rather complicated structure, however, we managed to provide the explicit formulas for this function. First, we solve the problem on an interval and then transfer our results to the circle and the line. We also obtain explicit estimates in multi-dimensional cases.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We find the best possible constant $C$ in the inequality $$\|\varphi\|_{L^r}^{\phantom{\frac{p}{r}}}\leq C\|\varphi\|_{L^p}^{\frac{p}{r}}\|\varphi\|_{\mathrm{BMO}}^{1-\frac{p}{r}}$$ for all possible values of parameters $p$ and $r$ such that $1 \le p < r < +\infty$. | result 0.50 | 0.96 | 0.84 | asserted 1.00 |
| s1 | We employ the Bellman function technique to solve this problem. | method 1.00 | 0.95 | 0.03 | asserted 1.00 |
| s2 | The Bellman function of three variables corresponding to this problem has a rather complicated structure, however, we managed to provide the explicit formulas for this function. | method 0.43 | 0.96 | 0.06 | asserted 1.00 |
| s3 | First, we solve the problem on an interval and then transfer our results to the circle and the line. | method 0.99 | 0.96 | 0.03 | asserted 1.00 |
| s4 | We also obtain explicit estimates in multi-dimensional cases. | result 0.68 | 0.96 | 0.08 | asserted 1.00 |
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Original abstract
We find the best possible constant $C$ in the inequality $$\|\varphi\|_{L^r}^{\phantom{\frac{p}{r}}}\leq C\|\varphi\|_{L^p}^{\frac{p}{r}}\|\varphi\|_{\mathrm{BMO}}^{1-\frac{p}{r}}$$ for all possible values of parameters $p$ and $r$ such that $1 \le p < r < +\infty$. We employ the Bellman function technique to solve this problem. The Bellman function of three variables corresponding to this problem has a rather complicated structure, however, we managed to provide the explicit formulas for this function. First, we solve the problem on an interval and then transfer our results to the circle and the line. We also obtain explicit estimates in multi-dimensional cases.
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How the literature uses it
1 arXiv paper cite this in 2 sentences
What it cites · 17 sentences without a confident label
Builds on 1 paper
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To prove Theorem [ref], we only need to deal with the function $B_2$, because the same theorem concerning $B_1$ was proved in [ref] .
Sharp mutliplicative inequalities with $\mathrm{BMO}$ $\mathrm{I}$ · Proof of Theorem [ref] · 0.88
Uses the method or tool of 3 papers
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We use Lemma 6.4 of [ref] to prove the sharpness of [ref] (and thereby Theorem [ref]).
Sharp mutliplicative inequalities with $\mathrm{BMO}$ $\mathrm{I}$ · On the circle and on the line: proofs of Theorems [ref] and [ref] · 0.96
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Therefore, by Corollary 5.1.6 from [ref] we obtain [equation]
Bellman function for extremal problems in $\mathrm{BMO}$ II: evolution · Optimizers for points in $ΞR+$ · 0.80
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Following [ref] , we prove dimension-free estimate using the Garcia-type norm on $(ℝ^n)$.
Dimension-free estimates for semigroup BMO and $A_p$ · Introduction · 0.95
Cites as background 1 paper
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All of that may be found in [ref] .
Sharp mutliplicative inequalities with $\mathrm{BMO}$ $\mathrm{I}$ · Introduction · 0.99
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